The goal of this section is to state a version of the Morse lemma in the Lie groupoid setting. Our approach relies on the nice geometric proof of existence of groupoid tubular neighborhoods around an orbit due to Meinrenken in [1]} combined with his ideas used to give a proof of the Morse-Bott Lemma. A different approach that may also be used to achieve a groupoid version of the Morse lemma is the classical one provided in [2]}. Recall that given a Morse-Bott function \(f:M\rightarrow \mathbb {R}\)  and a non-degenerate critical submanifold \(C\subseteq M\) , the Morse-Bott Lemma gives a local normal form for \(f\)  around \(C\) . Namely, on a suitable neighborhood of \(C\)  the function \(f\)  looks like the quadratic form \(Q_f:\nu (C)\rightarrow \mathbb {R}\)  defined in (REF ) up to a constant, see for instance [3]}.
